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Mirrors > Home > MPE Home > Th. List > Mathboxes > trpredex | Structured version Visualization version GIF version |
Description: The transitive predecessors of a relation form a set (NOTE: this is the first theorem in the transitive predecessor series that requires infinity). (Contributed by Scott Fenton, 18-Feb-2011.) |
Ref | Expression |
---|---|
trpredex | ⊢ TrPred(𝑅, 𝐴, 𝑋) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-trpred 31842 | . 2 ⊢ TrPred(𝑅, 𝐴, 𝑋) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) | |
2 | frfnom 7575 | . . . . 5 ⊢ (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) Fn ω | |
3 | omex 8578 | . . . . 5 ⊢ ω ∈ V | |
4 | fnex 6522 | . . . . 5 ⊢ (((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) Fn ω ∧ ω ∈ V) → (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) ∈ V) | |
5 | 2, 3, 4 | mp2an 708 | . . . 4 ⊢ (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) ∈ V |
6 | 5 | rnex 7142 | . . 3 ⊢ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) ∈ V |
7 | 6 | uniex 6995 | . 2 ⊢ ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) ∈ V |
8 | 1, 7 | eqeltri 2726 | 1 ⊢ TrPred(𝑅, 𝐴, 𝑋) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2030 Vcvv 3231 ∪ cuni 4468 ∪ ciun 4552 ↦ cmpt 4762 ran crn 5144 ↾ cres 5145 Predcpred 5717 Fn wfn 5921 ωcom 7107 reccrdg 7550 TrPredctrpred 31841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-trpred 31842 |
This theorem is referenced by: frmin 31867 |
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